OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 369 



equal volume fki is similarly concentrated ; this will have the effect 

 of moving the common centre of gravity of the system from n to z, 

 in a direction parallel to yt, and the distance nz, through which the 

 common centre is moved, is what the proposition determines. 



Let the position of the point z be supposed known ; then, if a vertical 

 line be drawn through that point perpendicular to the line of floatation, 

 the centre of gravity of the immersed space hied, will occur in some 

 point of that line, as for example at g ; but we have already observed, 

 that it is not necessary to determine the absolute position of the point 

 in question, the horizontal distance GS or rz between the verticals 

 Gr and mz, being all that is required. 



Put a zz: hied or efcd, the area of the immersed space, 



d z= hke or fki, the area of the triangle which has been 

 assumed as constituting an individual body of the 

 system ; 



d zzr yt, the horizontal distance through which the centre of 

 gravity of the triangle hke has moved, in shifting to the 

 position o in the triangle fki, 



S zz: Gg or Gn, the distance between the centre of effort and 

 the centre of buoyancy, when the axis of the section is 

 vertical ; 



b zz: A B or a b, the length of the greater parallel side of the 

 trapezoidal section, 



/3 zz: DC or dc, the length of the lesser parallel side ; 



D zz: PQ or pq, the perpendicular distance between the paral- 

 lels AB and DC, or ab and dc, 



c zz: EF or ef, the water line or line of floatation in the 

 upright position, 



I zz: the axis of motion, or the whole length of the floating 

 body, passing through o the centre of effort ; 



s zz: the specific gravity of the floating body, 



s' zz: the specific gravity of the supporting fluid, which in the 

 case of water, is expressed by unity ; 



5 zz: the stability of the body, or the momentum of the redress- 

 ing force ; 



< zzr/Az, or nGr, the angle of deflexion, and 



x zz: GS or rz, the length of the equilibrating lever. 



Then, by substituting the literal representatives of the several quan- 

 tities in the foregoing analogy, we shall obtain 



a : a' : : d: nz, f*\* > 



VOL. i. 2 B 



UN1VERS! 



